Backgrounds of arithmetic and geometry : an introduction / Radu Miron, Dan Branzei

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Chapter I presents the naive set theory with the constructions and structurings of the families of cardinal and ordinal numbers. Chapter II presents the natural numbers by means of the Frege-Russell constructions, Chapter III presents the axiomatic Theories differentiated according to their degree of formalization and the central metatheoretical problems. Chapter IV presents the algebraic backgrounds of Geometry. Chapter V edifies Geometry on the basis of Hilbert's axiomatic system. In chapter VI, Hilbert's axiomatics is correlated to that of Birkhoff, Chapter VII presents in brief the mean types of geometrical transformations, their actions upon the fundamental geometric figures and the results of certain geometrical transformations compounding. Chapter VIII exposes Felix Klein's conception on Geometries and exemplifies it through the study of affine, projective and plane hyperbolic Geometries. The last chapter presents the construction of Geometry through an axiomatics of Bachmann type, with Isometry as a primary notion. The first eight chapters contain exercises or problems.